↑

Properties

Label 2-507-13.3-c1-0-15
Degree $2$
Conductor $507$
Sign $-0.978 - 0.207i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 − 2.33i)2-s + (0.5 + 0.866i)3-s + (−2.62 + 4.54i)4-s − 1.04·5-s + (1.34 − 2.33i)6-s + (−0.277 + 0.480i)7-s + 8.74·8-s + (−0.499 + 0.866i)9-s + (1.41 + 2.44i)10-s + (−1.45 − 2.52i)11-s − 5.24·12-s + 1.49·14-s + (−0.524 − 0.908i)15-s + (−6.51 − 11.2i)16-s + (2.42 − 4.20i)17-s + 2.69·18-s + ⋯
L(s)  = 1  + (−0.951 − 1.64i)2-s + (0.288 + 0.499i)3-s + (−1.31 + 2.27i)4-s − 0.469·5-s + (0.549 − 0.951i)6-s + (−0.104 + 0.181i)7-s + 3.09·8-s + (−0.166 + 0.288i)9-s + (0.446 + 0.773i)10-s + (−0.438 − 0.760i)11-s − 1.51·12-s + 0.399·14-s + (−0.135 − 0.234i)15-s + (−1.62 − 2.82i)16-s + (0.588 − 1.01i)17-s + 0.634·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.978 - 0.207i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.978 - 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0414340 + 0.395249i\)
\(L(\frac12)\) \(\approx\) \(0.0414340 + 0.395249i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (1.34 + 2.33i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 1.04T + 5T^{2} \)
7 \( 1 + (0.277 - 0.480i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.45 + 2.52i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.42 + 4.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.376 + 0.652i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.88 + 4.99i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.955 - 1.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.51T + 31T^{2} \)
37 \( 1 + (2.87 + 4.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.45 + 4.25i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.54 + 9.61i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.753T + 47T^{2} \)
53 \( 1 + 7.58T + 53T^{2} \)
59 \( 1 + (2.04 - 3.54i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.71 + 2.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.936 - 1.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.25 + 9.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 - 1.33T + 79T^{2} \)
83 \( 1 - 2.64T + 83T^{2} \)
89 \( 1 + (4.96 + 8.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.53 - 14.7i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.64128788831642209250212212872, −9.571420339428290982751774812013, −8.988197377492536534008718441522, −8.151317967077337203638948180096, −7.36973325548153591504661722979, −5.36728774533997281546360926195, −4.04754829986213035827527160043, −3.26652226411625745720166923703, −2.22459711538792968601592361682, −0.31882335585537865144315791245, 1.59057039576891750322497331635, 3.90527510356296916419537629423, 5.25922960322207508350869261169, 6.15222130218542290734005154320, 7.11188544219701217831361820121, 7.83965806289024141021091723371, 8.229959762239122556067876943382, 9.466978069389482839475233341262, 9.977599315406989747278089846126, 11.09238626915138409147800085398

Graph of the $Z$-function along the critical line